# cold_plasma_permittivity_LRP¶

plasmapy.formulary.dielectric.cold_plasma_permittivity_LRP(B: Unit(‘T’), species, n, omega: Unit(‘rad / s’))

Magnetized cold plasma dielectric permittivity tensor elements.

Elements (L, R, P) are given in the “rotating” basis, i.e. in the basis $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$, where the tensor is diagonal and with $$B ∥ z$$.

The $$\exp(-i ω t)$$ time-harmonic convention is assumed.

Parameters
Returns

Notes

In the rotating frame defined by $$(\mathbf{u}_{+}, \mathbf{u}_{-}, \mathbf{u}_z)$$ with $$\mathbf{u}_{\pm}=(\mathbf{u}_x \pm \mathbf{u}_y)/\sqrt{2}$$, the dielectric tensor takes a diagonal form with elements L, R, P with:

\begin{align}\begin{aligned}L = 1 - \sum_s \frac{ω_{p,s}^2}{ω\left(ω - Ω_{c,s}\right)}\\R = 1 - \sum_s \frac{ω_{p,s}^2}{ω\left(ω + Ω_{c,s}\right)}\\P = 1 - \sum_s \frac{ω_{p,s}^2}{ω^2}\end{aligned}\end{align}

where $$ω_{p,s}$$ is the plasma frequency and $$Ω_{c,s}$$ is the signed version of the cyclotron frequency for the species $$s$$.

References

• T.H. Stix, Waves in Plasma, 1992.

Examples

>>> from astropy import units as u
>>> from numpy import pi
>>> B = 2*u.T
>>> species = ['e', 'D+']
>>> n = [1e18*u.m**-3, 1e18*u.m**-3]
>>> omega = 3.7e9*(2*pi)*(u.rad/u.s)
>>> L, R, P = permittivity = cold_plasma_permittivity_LRP(B, species, n, omega)
>>> L
<Quantity 0.63333...>
>>> permittivity.left    # namedtuple-style access
<Quantity 0.63333...>
>>> R
<Quantity 1.41512...>
>>> P
<Quantity -4.8903...>